Set S Consists of all Prime Integers Less Than 10. If Two Numbers are GMAT Problem Solving

Set S consists of all prime integers less than 10. If two numbers are chosen form set S at random, what is the probability that the product of these numbers will be greater than the product of the numbers which were not chosen?

1. 1/2
2. 2/3
3. 1/2
4. 7/10
5. 4/5

“Set S consists of all prime integers less than 10. If two numbers are” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

It is given in the question that all prime numbers smaller than 10 are included in set S. What is the likelihood that the product of two numbers selected at random from set S will be higher than the product of the numbers not chosen?
S = {2,3,5,7}
The easiest way to answer this question is to understand that we pick half of the numbers (basically, we split the group of 4 into two smaller groups of 2), and since a tie is impossible, the probability that the product of the numbers in either of the two subgroups is greater than that of the other is 1/2 (the probability doesn't favour either of the two subgroups).

The correct answer is C.

Correct Answer: C

Approach Solution 2:

Prime numbers less than 10: 2,3,5,7 ⇒ 4 numbers in set S
2 are selected ==> 4!/2!*2!= Six different ways to choose two numbers from set S
Favorable outcomes: 3-5, 3-7, 5-7 ⇒ 3
P= 3/6 = 1/2

Hence, the correct answer is 1/2.

Correct Answer: C

Approach Solution 3:

Understanding that there cannot be a case in which the product of two picked integers equals the product of two left integers is crucial to solving this puzzle. The fact that every integer in the set is a prime number allows one to draw this conclusion.
There can be 6 different products,
Total outcome i.e. selecting 2 numbers out of 4 -> 4C2
For the product of selected numbers to be greater, neither number should be 2 as max product for 2 = 2*7 => 14 which is less than min product for next greater number 3 i.e. 3*5 => 15.
So both numbers should be 3,5 or 7.
So, the probability for selecting 2 numbers with greater product = 3C2 / 4C2 i.e.choosing two numbers from 3,5,7 divided by all possible choices equals half.

Correct Answer: C

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